Quantum Mechanics as Hamilton–Killing Flows on a Statistical Manifold

Abstract

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated.

Entropic Density Functional Theory

“…To summarize our conclusions: We have produced a reconstruction of DFT that makes explicit how DFT fits within an ongoing research program that places the concepts of entropy and information at the very foundation for all of physics (see, e.g., [34]). This includes statistical mechanics [22][23][24][25][26], quantum mechanics [35,43], and as we have shown in this work, also the main techniques to study structure-variational principles including mean field methods and DFT.…”

“…Information is just the constraints we decide to impose on our beliefs, and there is no need to define "amounts" of information. The motivation behind the design criteria was clarified and their number reduced from five to two [30][31][32][33] (reviewed in [34,35]).…”